{
  "id": "1412.8082",
  "version": "v1",
  "published": "2014-12-27T21:49:27.000Z",
  "updated": "2014-12-27T21:49:27.000Z",
  "title": "A rationality result for the exterior and the symmetric square $L$-function",
  "authors": [
    "Harald Grobner"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $G={\\rm GL}_{2n}$ over a totally real number field $F$ and $n\\geq 2$. Let $\\Pi$ be a cuspidal automorphic representation of $G(\\mathbb A)$, which is cohomological and a functorial lift from SO$(2n+1)$. The latter condition can be equivalently reformulated that the exterior square $L$-function of $\\Pi$ has a pole at $s=1$. In this paper, we prove a rationality result for the residue of the exterior square $L$-function at $s=1$ and also for the holomorphic value of the symmetric square $L$-function at $s=1$ attached to $\\Pi$. On the way, we also show a rationality result for the residue of the Rankin--Selberg $L$-function at $s=1$, which is very much in the spirit of our recent joint paper with Harris and Lapid, as well as of one of the main results in a recent article of Balasubramanyam--Raghuram.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-27T21:49:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rationality result",
      "symmetric square",
      "exterior square",
      "cuspidal automorphic representation",
      "totally real number field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}